Aspnet.Identity is way to authenticate users. It makes life simple for you, creating code for cookie generation, maintaining database relationship, password recovery, password encryption, email intergration, extends authentication to link with Twitter, Facebook, Google, etc. It is a lot more that "user authentication against registered users inside a table."
You can find more information on Identity here: http://www.asp.net/identity/overview/getting-started/introduction-to-aspnet-identity
Microsoft has been continuously improving Identity making it more secure. However, there are always ways and one cannot help if the developer commits mistake.
However, given all the updates- Identity code has changed so much so that solutions which were valid 1 year back are no longer valid. One needs to dig deep to make simple changes for customization.
Here is how you can integrate Identity with a custom storage provider like MySQL: http://www.asp.net/identity/overview/getting-started/aspnet-identity-using-mysql-storage-with-an-entityframework-mysql-provider
Please note that primary key by default is GUID and currently there is no easy way to change the primary for MySQL integration to INT. You will find more details here: https://forums.asp.net/t/2062668.aspx?How+to+change+primary+key+for+Identity+with+MySQL+as+storage+provider
Moreover, there are many issues with MySQL and LINQ integration. Lazy loading issues, etc. This would be very evident even with a minor customization of Identity code. If you want to include Identity for user authentication I strongly suggest shifting to MS SQL instead of MySQL